This paper deals with accessibility of kinematic asymmetric affine control systems with affine constraints (KAACS). We first show that the KAACS can be uniquely derived from affine constraints and control inputs. We next investigate relations between accessible sets of the KAACS and foliation structures in the cases of complete integrability, partial nonintegrability and complete nonintegrability. Then, necessary and sufficient conditions for local accessibility and strong local accessibility of the KAACS are derived. Finally, we show some examples in physical and engineering systems and apply our results to them.